evaluate without using a calculator or mathematical table:log32+log128-log729/log32+log2-log27
To evaluate the expression without a calculator or mathematical table, we can use the properties of logarithms.
First, let's simplify each logarithm:
1. log32 = log(2^5) = 5log2
2. log128 = log(2^7) = 7log2
3. log729 = log(3^6) = 6log3
4. log27 = log(3^3) = 3log3
5. log2 = log(2^1) = log2
Now, substitute these simplified logarithms back into the expression:
log32 + log128 - log729 / log32 + log2 - log27
= 5log2 + 7log2 - 6log3 / 5log2 + log2 - 3log3
Using the properties of logarithms, we can combine the logarithms with the same base:
= (5 + 7)log2 - (6 + 3)log3 / (5 + 1)log2 - 3log3
= 12log2 - 9log3 / 6log2 - 3log3
Now, let's simplify the expression further:
= 3(4log2 - 3log3) / 3(2log2 - log3)
= 4log2 - 3log3 / 2log2 - log3
Since the denominators are the same, we can treat this expression as a fraction:
= (4log2 - 3log3) / (2log2 - log3)
Now, let's evaluate the expression without a calculator or mathematical table:
We can rewrite each logarithm using the change of base rule:
= (4log(10)/log(2) - 3log(10)/log(3)) / (2log(10)/log(2) - log(10)/log(3))
Since log(10) is a commonly known logarithm value, we can substitute it with its approximate value of 1:
= (4(1)/log(2) - 3(1)/log(3)) / (2(1)/log(2) - (1)/log(3))
= (4/log(2) - 3/log(3)) / (2/log(2) - 1/log(3))
Now, we need to approximate the logarithmic values:
- log(2) is approximately 0.301
- log(3) is approximately 0.477
Substituting these values into the expression:
= (4/0.301 - 3/0.477) / (2/0.301 - 1/0.477)
= (13.29 - 6.29) / (6.63 - 2.09)
= 7 / 4.54
= 1.541
Therefore, the approximate value of the expression "log32+log128-log729/log32+log2-log27" is approximately 1.541.
To evaluate the expression without using a calculator or mathematical table, we can simplify the given logarithmic expressions using the properties of logarithms.
Let's break down the expression step-by-step:
1. log32 + log128:
Using the property log(a) + log(b) = log(ab), we can simplify this to log(32 x 128).
2. log(32 x 128):
To simplify this further, we can evaluate 32 x 128:
32 x 128 = 4096.
So, log(32 x 128) = log(4096).
3. log(4096):
To evaluate this logarithm without a calculator, we can express 4096 as a power of 2 since the logarithm is in base 2.
2^12 = 4096.
Therefore, log(4096) = log(2^12) = 12.
Now, let's continue evaluating the second part of the expression:
4. log729 / log32:
Using the property log(a) - log(b) = log(a/b), we can simplify this expression to log(729/32).
5. log(729/32):
To evaluate this logarithm, we need to express 729 and 32 as powers of the same base.
729 = 9 x 9 x 9 = 3^6.
32 = 2^5.
So, log(729/32) = log(3^6/2^5).
6. log(3^6/2^5):
Applying the property log(a/b) = log(a) - log(b), we can simplify this to 6log(3) - 5log(2).
Now, let's evaluate the final part of the expression:
7. log27:
To evaluate this logarithm without a calculator, we can express 27 as a power of 3 since the logarithm is in base 3.
3^3 = 27.
Therefore, log27 = log(3^3) = 3.
Now, let's put it all together:
log32 + log128 - log729/log32 + log2 - log27 = 12 - (6log(3) - 5log(2))/3
We cannot simplify this expression any further without using a calculator.